Solving the Quispel-Roberts-Thompson maps using Kajiwara-Noumi-Yamada's representation of elliptic curves

Abstract

It is well known that the dynamical system determined by a Quispel-Roberts-Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on ${\mathbb{CP}}^1 \times {\mathbb{CP}}^1$. In most cases this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of genus one, and the system can be solved as a translation on the elliptic fiber to which the initial point belongs. However, this procedure is rather complicated to handle, especially in the normalization process. In this paper, for a given initial point on an invariant elliptic curve, we present a method to construct the solution directly in terms of the Weierstrass sigma function, using Kajiwara-Noumi-Yamada's parametric representation of elliptic curves.